Unwind di Power Electronics - 1 di 5

Lesson #01

Lesson #02

The buck converter is composed by a couple of ideal switches and a DC voltage. The switches are pilotated by PWM signals. The circuit is:

Lesson #03

OFFS ON sVg isch s e 0 son OFFVg Altiietti 5The signals c(t) and c’(t) are generated using Pulse Width Modulator (PWM):

COMPARATORCUIUNOral citiOSCILLATOR

The output voltage is a Pulse Width modulated signal and the waveforms of u(t) (that in general could be a constant “U”), vr(t) (that is the signal generated by the oscillator), c(t) and vx(t) are:

Viotti Vr tTsciti “Ts” is called “switching period”.

The average value Vx of the output waveform is related to the duty cycle as: Vx IvgVgD VxE DI 0 I0 1

The maximum value of the output is the value of the DC input, so for this reason, the buck converter is called also “step-down” converter. Adjusting the duty cycle it can be adjusted the average value of the output voltage. In terms of average value, these results are good, but instantaneous value of output waveform is not constant, so output waveform is not a constant, in fact is a square waveform so has a lot of harmonic contents.

Due to the harmonic contents vx(t), the circuit needs a low pass filter to filter out the harmonic contents generated by switches S and S’.

The simplest way to filter out this harmonic content and take constant the DC component of vx(t) it can be used an RC filter:

CHI s RVg Alti e ric14 5

In this case, R and C can be properly designed to filter out the harmonic content. This solution is not very good due to the fact that R will dissipate energy and the current will flow through R when switch S is close.

A better filter is a second order filter composed by an inductor and a capacitor:

Buck converter CHI s LVg Alti e Riioc14 5

In this circuit, named “buck converter” can be finally drawn the voltage vo. Let’s analyse now the spectrum of Vx:

vitti e fGfsZfsfs Sfs fs50

The better position of the filtering frequency is before “fs” but it can be done in different ways, as shown here. Let’s derive now the transfer function of the filter:

s'eEHIS iislet 52SL Le iI t adEWo iLe

The location of the frequency of the filter is more or less this:

oi.tl It’s not a Bode diagram! Frequency axis is linear! lo 4Cfsfs fs fs fs5O 3

The lower is f0, the largest will be L and C. On the other hand, L and C must not be too much small due to fact that f0 must not be closer to “fs” because the filter action could not be good. Usually the frequency of the filter is imposed at 1 decade before the switching frequency:

fot

The general expression of vx() that presents the superposition of DC and AC components is: II VgiVxtilt eoslkwsti.lk Vx Ddate

The expression of vo(t) takes count of the action of the filter and it is:

HIIKWDGslkwsttl.it014 it'ÈJolt It an It'kUSD1

Where “H(j*0)” is the magnitude of the filter calculated at DC frequency and looking to previous expression it’s simply equal to 1: H(j*0)=1. For this reason: H(j*0)*Vx=D*Vg=Vo.

The expression shows how the filter affects every single harmonic, including DC. Assuming an opportune design of the filter, the cut-off frequency of the filter is much more small then the switching frequency: Wo Us LlLL 1

Considering this hypothesis, the expression of the modulus of the response of the filter becomes:

ÈIt'skins E l1 1ate iii KYIwe WsLLJk 1Ws5 a LL 1

The previous expression of vo(t) becomes: 2 Gslkwsttlxtttlir.wsDivertìtolti a II

The important result is that the amplitude of each harmonic is strongly attenuated due to the fact that there is the factor “k” at the denominator and due to the fact that: (w0/ws)^2, is a very small quantity. The AC component of vo(t) becomes so very small and is named “switching ripple” and it can NEVER be completely filtered out because there is not filter they can infinitely attenuate the harmonic components.

The output vo(t) is so: Dtolti Small ita rippleVg

An appropriate waveform of vo(t) is so: vvoltiVoting0 EThe output voltage slowly oscillates around the desired DC voltage.

Voltage conversion ratio

Let’s define now the voltage conversion ratio of the converter. The voltage conversion ratio of the converter is indicated with “M” and is defined as the ratio between the average output voltage and the input voltage:

MI di The buck converter is a stepe di.ve down converter because output voltage cannot exceed the input voltage MBUCK D14I Di

Let’s try now to sketch the waveform of the inductor. To analyse this waveform let’s consider the previous circuit:

IctCHI s illaVg LAlti e Riol'Itt s

The voltage across the inductor is a function of time and changes during “Ton” and “Toff”, in particular:

t Tauvice a iJo Tafft

This is the exact expression of the voltage across the inductor as a function of time. The expression of the voltage of the inductor is also: Vg Jolt TONLriti a Iddio e aaddii Toret

The waveform assumed by the current is similar to a triangle but due to the ripple superposed over the output voltage, the current waveform is not a precise triangle. Anyway, as long as the output voltage ripple is small: vo(t)=Vo, so: Vg Jolt Ton0adit E Lott LO Torefoce

Remembering that Vo cannot exceed Vg, the slope of the waveform of the current is positive (the current increases linearly) during Ton and negative (the current decreases linearly) during Toff.

For these reasons, the waveform of the current in the inductor is: U or ItVr tTscitiI 5 ON S OFF0 EOHHVg tiltIlIL t

The average value of the current in the inductor corresponds to the current absorbed by the load:

IctCHI s E IoiettiVg LAlti e io Ril'Itt s

The current Io can be considered approximately constant due to the fact that the output voltage is approximately constant. As the output contains ripple, also the output current contains ripple.

Another interesting plot is the input current. The input current is equal to the inductor current during “Ton” and 0 during “Toff” because S and S’ are respectively closed/opened and opened/closed.

So the current in the input DC voltage is: id IltlLL Liotti E It’s clear that the input current is not constant. So normally there are input filters too.

The simplest way to filter out the harmonic content of the input current is given by a capacitor put in parallel to the input DC voltage:

IctCHI s illaCgVg Ltt ld Rtol'Itt s

Let’s suppose to calculate the variation of the inductor current during “Ton”:

inIEI 1 il tTtI Ton

This quantity of current is named “peak-to-peak inductor current ripple” and is given by: dire didi rittida

This is the exact expression, but it can be applied an approximation. In fact, vL(t) can be approximated to a constant during “Ton” (also during “Toff”): è dit E ivs vo Vovi Ton0ITL Ton Slope of iL(t) during “Ton”.

This last result is based on the assumption that the voltage across the inductor is constant, but this means that it has been applied the small ripple approximation:

E VoJolt Iolle avg.lvdictidt Ott ETON

The small ripple approximation is a very useful approximation but usually, it cannot be applied to every converter.

Steady-state analysis

A more general way to analyse the converters is the so-called “steady-state” analysis of the converter. The expression “steady-state” is synonymous with “periodic”. In this particular analysis, all the inputs are considered constants (input DC voltage and duty cycle D are constants). Moreover, in steady-state conditions, all transistors are extinguished and all the voltages and all the currents become periodic with a common fundamental period that is the switching period “Ts”. All the waveforms have the same period and it’s equal to the switching period. So, when a converter is in “steady-state”, all waveforms are periodic:

ictOlt iiltt ttTs Tsa e

Obviously, if a converter goes out of steady-state conditions, it means that the waveforms become a-periodic. This result has important consequences that are 2 important results in converter analysis. Using steady-state hypothesis, it can be developed a couple of theorems that are “the volt-second balance principle” and “the ampere-second balance principle”.

Let’s see firstly the volt-second balance principle considering an ideal inductor:

didtilt HLA li